Homework 7: State

Due: 04/12 at 11:55pm
Points: 15 (each question 5 points)

Please prepare your solutions as a PDF file (scanned handwriting is okay, but no .doc or .docx, please). Submit your PDF file in Sakai.

1. Call-by-reference

In class, we defined the following transformation from Scheme-style mutation to ML-style:

     T⟦x⟧ = !x
 T⟦λx. t⟧ = λx. let x = ref x in T⟦t⟧
 T⟦t₁ t₂⟧ = T⟦t₁⟧ T⟦t₂⟧
T⟦x := t⟧ = x := T⟦t⟧

Some languages allow call-by-reference: in C++, this is indicated by putting & in front of the name of the formal parameter. In the lambda calculus, we could introduce new syntax

t ::= ...
      &x. t
      t @ x

where & is like λ but uses call-by-reference, and t @ x applies a call-by-reference function to x, so that

let inc = &x. x:=succ x in
let x = 0 in
  inc @ x;
  x
⟶* succ 0

How would you extend the above transformation to handle this?

 T⟦&x. t⟧ = 
 T⟦t @ x⟧ =

The right-hand sides should use only simply-typed or untyped (your choice) λ-calculus with references.

Hint: The answer is not very complicated, so don’t worry if it looks too simple.

2. Mutable lists

Assume simply-typed λNB with ~~NatList (cons, car, cdr)~~ recursive types (μ) and references (ref, !, and :=).

In Scheme, set! corresponds roughly to :=. It also has set-car! and set-cdr!, which can be used to modify lists:

> (define l (list 1 2 3))
> (set-car! l 4)
> l
(4 2 3)
> (set-cdr! l (list 5 6))
> l
(4 5 6)

Define a type MutableNatList and associated functions:

  nil' : MutableNatList
 cons' : Nat → MutableNatList → MutableNatList
  car' : MutableNatList → Nat
  cdr' : MutableNatList → MutableNatList
setcar : MutableNatList → Nat → Unit
setcdr : MutableNatList → MutableNatList → Unit

They should satisfy the following equations:

car' nil'        = 0
car' (cons' x y) = x
cdr' nil'        = nil'
cdr' (cons' x y) = y
(let l = cons' x y in setcar l x'; car' l) = x'
(let l = cons' x y in setcar l x'; cdr' l) = y
(let l = cons' x y in setcdr l y'; car' l) = x
(let l = cons' x y in setcdr l y'; cdr' l) = y'

Show how to build a circular list of zeros such that the following equations are true:
```
car' zeros = 0
cdr' zeros = zeros
```

3. Recursion via references

Assume simply-typed λNB with multiplication (times) and references (ref, !, and :=).

Define a factorial function fac : Nat→Nat. Hint: A function that calls itself is not so different from a list that contains itself as a tail (1c 2b).
Define fix : ((Nat→Nat) → Nat→Nat) → Nat→Nat. It should be possible to use fix to define a factorial function in the usual way:
```
fac ≡ fix (λf: Nat→Nat. λn:Nat. if iszero n then succ 0 else times n (f (pred n)))
```